Trapping Horizons in the Sultana-Dyer Space-Time

The Sultana-Dyer space-time is suggested as a model describing a black hole embedded in an expanding universe. Recently, in \cite{0705.4012}, its global structure is analyzed and the trapping horizons are shown. In the paper, by directly calculating the expansions of the radial null vector fields normal to the space-like two-spheres foliating the trapping horizons, we find that the trapping horizon outside the event horizon in the Sultana-Dyer space-time is a past trapping horizon. Further, we find that the past trapping horizon is an outer, instantaneously degenerate or inner trapping horizon accordingly when the radial coordinate is less than, equal to or greater than some value.